Zeta and L functions of Voevodsky motives
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913607931920384 |
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| author | Kahn, Bruno |
| author_facet | Kahn, Bruno |
| contents | We associate an $L$-function $L^{\mathrm{near}}(M,s)$ to any geometric motive over a global field $K$ in the sense of Voevodsky. This is a Dirichlet series which converges in some half-plane and has an Euler product factorisation. When $M$ is the dual of $M(X)$ for $X$ a smooth projective variety, $L^{\mathrm{near}}(M,s)$ differs from the alternating product of the zeta functions defined by Serre in 1969 only at places of bad reduction; in exchange, it is multiplicative with respect to exact triangles. If $K$ is a function field over $\mathbf{F}_q$, $L{\mathrm{near}}(M,s)$ is a rational function in $q^{-s}$ and enjoys a functional equation. The techniques use the full force of Ayoub's six (and even seven) operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeta and L functions of Voevodsky motives Kahn, Bruno Number Theory Algebraic Geometry We associate an $L$-function $L^{\mathrm{near}}(M,s)$ to any geometric motive over a global field $K$ in the sense of Voevodsky. This is a Dirichlet series which converges in some half-plane and has an Euler product factorisation. When $M$ is the dual of $M(X)$ for $X$ a smooth projective variety, $L^{\mathrm{near}}(M,s)$ differs from the alternating product of the zeta functions defined by Serre in 1969 only at places of bad reduction; in exchange, it is multiplicative with respect to exact triangles. If $K$ is a function field over $\mathbf{F}_q$, $L{\mathrm{near}}(M,s)$ is a rational function in $q^{-s}$ and enjoys a functional equation. The techniques use the full force of Ayoub's six (and even seven) operations. |
| title | Zeta and L functions of Voevodsky motives |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2412.08437 |